Question

A solid, homogeneous sphere of mass M and radius R is rolling down a rough inclined plane inclined at an angle \(\theta\). Which of the following statements is INCORRECT?

• A. Angular momentum is not conserved about the point of contact.
• B. The net torque acting on the sphere is \(MgR\cos \theta\)
• C. The total energy is conserved.
• D. The component of force on the sphere along the incline is \(Mg\sin \theta -f=Ma\)

Hint:

Compare each statement with the torque about the centre and the contact point, and with the energy of pure rolling.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
For rolling without slipping, the contact point is momentarily at rest, so static friction does no work. Torque depends on the point about which it is calculated.

Step 2: Check the true statements
Angular momentum about the contact point is not conserved, because the weight has a non-zero torque about it. Total mechanical energy is conserved because friction is static. Along the incline, \(Mg\sin\theta-f=Ma\) is the correct equation of motion.

Step 3: Check the torque statement
About the centre, only friction acts, giving torque \(fR\) with \(f=\dfrac{2}{7}Mg\sin\theta\). About the contact point, the torque is \(MgR\sin\theta\). A torque of \(MgR\cos\theta\) does not arise about either point.

Final Answer:
The statement about net torque \(MgR\cos\theta\) is incorrect. \[ \boxed{\text{(B) }\text{The net torque acting on the sphere is }MgR\cos \theta} \]